"Hun skjønner ikke hva jeg sier."

Translation: She does not understand what I am saying.

[larlyssa]

What's the difference between å skjønne and å forstå?

[Regney]

"Realise" is often a good translation for skjønne. Another translation for skjønne is something little kids say: "I don't get it!"

Å skjønne is related to that which is apparent (based on perception). Å forstå is related to that which is factual (can be proven).
The difference can be illustrated in the reading of a document in unfamiliar parlance (e.g., a high school student reading a legal brief), "I realise what it says, but I don't understand what it means," (Jeg skjønner det som står der, men jeg forstår ikke hva det betyr.)

[fveldig]

Most of the time they're interchangeable.

[tengg.peter]

Jeg skjønner ikke hva hun sier... Jeg måtte lytte den sakte versjonen.

[astralcalculus]

Would it also be correct to say "Hun skjønner ikke det jeg sier"?

[Regney]

Yes. That's correct, too. You can also say, "... ikke det som jeg sier."